3.1112 \(\int \frac{1}{(1-x)^{5/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=41 \[ \frac{\sqrt{x+1}}{3 \sqrt{1-x}}+\frac{\sqrt{x+1}}{3 (1-x)^{3/2}} \]

[Out]

Sqrt[1 + x]/(3*(1 - x)^(3/2)) + Sqrt[1 + x]/(3*Sqrt[1 - x])

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Rubi [A]  time = 0.0248556, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{x+1}}{3 \sqrt{1-x}}+\frac{\sqrt{x+1}}{3 (1-x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(5/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(3*(1 - x)^(3/2)) + Sqrt[1 + x]/(3*Sqrt[1 - x])

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Rubi in Sympy [A]  time = 3.56795, size = 29, normalized size = 0.71 \[ \frac{\sqrt{x + 1}}{3 \sqrt{- x + 1}} + \frac{\sqrt{x + 1}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(5/2)/(1+x)**(1/2),x)

[Out]

sqrt(x + 1)/(3*sqrt(-x + 1)) + sqrt(x + 1)/(3*(-x + 1)**(3/2))

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Mathematica [A]  time = 0.013542, size = 23, normalized size = 0.56 \[ -\frac{(x-2) \sqrt{1-x^2}}{3 (x-1)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(5/2)*Sqrt[1 + x]),x]

[Out]

-((-2 + x)*Sqrt[1 - x^2])/(3*(-1 + x)^2)

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Maple [A]  time = 0.006, size = 18, normalized size = 0.4 \[ -{\frac{-2+x}{3}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(5/2)/(1+x)^(1/2),x)

[Out]

-1/3*(1+x)^(1/2)*(-2+x)/(1-x)^(3/2)

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Maxima [A]  time = 1.49236, size = 51, normalized size = 1.24 \[ \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*(-x + 1)^(5/2)),x, algorithm="maxima")

[Out]

1/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 1/3*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.207634, size = 90, normalized size = 2.2 \[ \frac{x^{3} + 3 \, x^{2} - 3 \,{\left (x^{2} - 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x}{3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*(-x + 1)^(5/2)),x, algorithm="fricas")

[Out]

1/3*(x^3 + 3*x^2 - 3*(x^2 - 2*x)*sqrt(x + 1)*sqrt(-x + 1) - 6*x)/(x^3 - (x^2 - 3
*x + 2)*sqrt(x + 1)*sqrt(-x + 1) - 3*x + 2)

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Sympy [A]  time = 18.2424, size = 128, normalized size = 3.12 \[ \begin{cases} \frac{x + 1}{3 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{3}{3 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{i \left (x + 1\right )}{3 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{1 - \frac{2}{x + 1}}} + \frac{3 i}{3 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{1 - \frac{2}{x + 1}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x)**(5/2)/(1+x)**(1/2),x)

[Out]

Piecewise(((x + 1)/(3*sqrt(-1 + 2/(x + 1))*(x + 1) - 6*sqrt(-1 + 2/(x + 1))) - 3
/(3*sqrt(-1 + 2/(x + 1))*(x + 1) - 6*sqrt(-1 + 2/(x + 1))), 2*Abs(1/(x + 1)) > 1
), (-I*(x + 1)/(3*sqrt(1 - 2/(x + 1))*(x + 1) - 6*sqrt(1 - 2/(x + 1))) + 3*I/(3*
sqrt(1 - 2/(x + 1))*(x + 1) - 6*sqrt(1 - 2/(x + 1))), True))

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GIAC/XCAS [A]  time = 0.207463, size = 30, normalized size = 0.73 \[ -\frac{\sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*(-x + 1)^(5/2)),x, algorithm="giac")

[Out]

-1/3*sqrt(x + 1)*(x - 2)*sqrt(-x + 1)/(x - 1)^2